\[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx \]
Optimal antiderivative \[ -\frac {e^{\frac {3}{2}} \arctan \! \left (1-\frac {\sqrt {2}\, \sqrt {e x}}{\left (-x^{2}+1\right )^{\frac {1}{4}} \sqrt {e}}\right ) \sqrt {2}}{8}+\frac {e^{\frac {3}{2}} \arctan \! \left (1+\frac {\sqrt {2}\, \sqrt {e x}}{\left (-x^{2}+1\right )^{\frac {1}{4}} \sqrt {e}}\right ) \sqrt {2}}{8}-\frac {e^{\frac {3}{2}} \ln \! \left (\sqrt {e}-\frac {\sqrt {2}\, \sqrt {e x}}{\left (-x^{2}+1\right )^{\frac {1}{4}}}+\frac {x \sqrt {e}}{\sqrt {-x^{2}+1}}\right ) \sqrt {2}}{16}+\frac {e^{\frac {3}{2}} \ln \! \left (\sqrt {e}+\frac {\sqrt {2}\, \sqrt {e x}}{\left (-x^{2}+1\right )^{\frac {1}{4}}}+\frac {x \sqrt {e}}{\sqrt {-x^{2}+1}}\right ) \sqrt {2}}{16}-\frac {e \left (-x^{2}+1\right )^{\frac {3}{4}} \sqrt {e x}}{2} \]
command
integrate((e*x)**(3/2)/(1-x)**(1/4)/(1+x)**(1/4),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {i e^{\frac {3}{2}} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{8}, - \frac {1}{8} & - \frac {1}{2}, - \frac {1}{4}, 0, 1 \\-1, - \frac {5}{8}, - \frac {1}{2}, - \frac {1}{8}, 0, 0 & \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi \Gamma \left (\frac {1}{4}\right )} - \frac {e^{\frac {3}{2}} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {5}{4}, - \frac {9}{8}, - \frac {3}{4}, - \frac {5}{8}, - \frac {1}{4}, 1 & \\- \frac {9}{8}, - \frac {5}{8} & - \frac {5}{4}, -1, - \frac {3}{4}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi \Gamma \left (\frac {1}{4}\right )} \]